Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/101

 Rh from the ends. When κ is infinite the free magnetism at any point of the cylinder is simply proportional to its distance from the middle point, the distribution being- similar to that of free electricity on a conductor in a field of uniform force.

440.] In all substances except iron, nickel, and cobalt, the coefficient of magnetization is so small that the induced magnetization of the body produces only a very slight alteration of the forces in the magnetic field. We may therefore assume, as a first approximation, that the actual magnetic force within the body is the same as if the body had not been there. The superficial magnetization of the body is therefore, as a first approximation, $$-\kappa \frac{dV}{d\nu}$$, where $$\frac{dV}{d\nu}$$ is the rate of increase of the magnetic potential due to the external magnet along a normal to the surface drawn inwards. If we now calculate the potential due to this superficial distribution, we may use it in proceeding to a second approximation.

To find the mechanical energy due to the distribution of magnetism on this first approximation we must find the surface-integral taken over the whole surface of the body. Now we have shewn in Art. 100 that this is equal to the volume-integral taken through the whole space occupied by the body, or, if R is the resultant magnetic force,

Now since the work done by the magnetic force on the body during a displacement δx is Xδx where X is the mechanical force in the direction of x, and since which shews that the force acting on the body is as if every part of it tended to move from places where R2 is less to places where it is greater with a force which on every unit of volume is